Convergence to Equilibrium for Time-Inhomogeneous Jump Diffusions with State-Dependent Jump Intensity

We consider a time-inhomogeneous Markov process X=(Xt)t and we are interested in its longtime behavior. The infinitesimal generator of the process is given for any sufficiently smooth test function f by Ltf(x)= n-ary sumation i=1d partial differential f partial differential xi(x)bi(t,x); Rm[f(x+c(t,z,x))-f(x)]gamma(t,z,x)mu(dz), Moreover, we introduce a coupling method for the limit process which is entirely based on certain of its big jumps and which relies on the regeneration method. We state explicit conditions in terms of the coefficients of the process allowing control of the speed of convergence to equilibrium both for X and for X over bar (X) over bar.